Abstract nonsense
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Abstract nonsense is a popular term used by mathematicians to describe certain kinds of arguments and concepts in category theory. This term is believed to have been coined by the mathematician Norman Steenrod,[1][2] himself one of the developers of the categorical point of view. This term is used by practitioners as an indication of mathematical sophistication or coolness rather than as a derogatory designation.[3]
Certain ideas and constructions in mathematics display a uniformity throughout many domains. The unifying theme is category theory. Rather than enter an elaborate discussion on particulars of arguments, mathematicians will use the expression Such and such is true by abstract nonsense. Typical instances are arguments involving diagram chasing, application of the definition of universal property, definition of natural transformations between functors, use of the Yoneda lemma and so on.
Other, possibly less flattering, characterizations of abstract reasoning have been recorded, although they have not acquired the status of mathematical jargon. For example, Paul Albert Gordan described a proof of David Hilbert's in invariant theory, saying "This is not mathematics; this is theology."[4]
[edit] Notes and references
- ^ Colin McLarty, The Uses and Abuses of the History of Topos Theory, Brit. J. Phil. Sci, 41 (1990) p 355.
- "[Steenrod] jokingly tagged category theory 'abstract nonsense' and made it central to his axiomatics for homology"
- ^ Joseph Rotman, Review of "An Introduction to Homological Algebra", Bull. Amer. Math. Soc., 33, (1995), p 475
- ^ Michael Monastyrsky, Some Trends in Modern Mathematics and the Fields Medal Can. Math. Soc. Notes, March and April 2001, Volume 33, nos. 2 and 3. Online version available [1].
- in algebra, the term “abstract nonsense” has a definite meaning without any pejorative connotation
- ^ Hermann Weyl, David Hilbert. 1862-1943, Obituary Notices of Fellows of the Royal Society (1944). The proof in question was the (non-constructive) existence of a finite basis for invariants. This proof seemed to counter the sensibilities of Gordan, whom Weyl described in his Hilbert obituary as a great "algorithmician."
